intersections and compact sets

Show that if K is compact and F is closed, then K intersect F is compact.

Alright this is what I have so far.

Suppose K is compact, which implies that K is closed and bounded. Since F is closed and K is closed, by theorem that means that K intersect F is closed as well. Since K intersect F is closed, that means that every cauchy sequence in K int. F has a limit that is an element of K int. F.

This is where I get stuck, I know that the definition of a compact set is:
A set K in R is compact if every sequence in K has a sub sequence that converges to a limit that is also in K.

I feel like that is what I concluded in my proof, so I could just say that means that K int. F is compact, however, it really isn't saying anything about sub sequences. Any help would be great!!!

Show that if K is compact and F is closed, then K intersect F is compact.

Alright this is what I have so far.

Suppose K is compact, which implies that K is closed and bounded. Since F is closed and K is closed, by theorem that means that K intersect F is closed as well. Since K intersect F is closed, that means that every cauchy sequence in K int. F has a limit that is an element of K int. F.

This is where I get stuck, I know that the definition of a compact set is:
A set K in R is compact if every sequence in K has a sub sequence that converges to a limit that is also in K.

I feel like that is what I concluded in my proof, so I could just say that means that K int. F is compact, however, it really isn't saying anything about sub sequences. Any help would be great!!!

If you're working in , then we can use the Bolzano-Weierstrass theorem which states that is sequentially compact if and only if it is closed and bounded.